MEV442 Introduction to Robotics Module 2. Dr. Santhakumar Mohan Assistant Professor Mechanical Engineering National Institute of Technology Calicut

Transcription

1 MEV442 Introducton to Robotcs Module 2 Dr. Santhakumar Mohan Assstant Professor Mechancal Engneerng Natonal Insttute of Technology Calcut

2 Jacobans: Veloctes and statc forces Introducton Notaton for tme-varyng poston and orentaton Lnear and rotatonal velocty of rgd bodes More on angular velocty Moton of the lnks of a robot Velocty propagaton from lnk to lnk Jacobans Sngulartes Statc forces n manpulator Jacobans n the force doman Cartesan transformaton of veloctes and statc forces 1

3 Introducton In ths chapter, we expand our consderaton of robot manpulators beyond statc poston problems. We examne the notons of lnear and angular velocty of a rgd body and use these concepts to analyze the moton of a manpulator. We also wll consder force actng on a rgd body, and then use these deas to study the applcaton of statc force wth manpulators. It turns out that the study of both velocty and statc forces leads to a matrx entty called the Jacoban of the manpulator, whch wll be ntroduced n ths chapter. 2

4 Central Topc - Smultaneous Lnear and Rotatonal Velocty Vector Form Matrx Form 3

5 Defntons - Lnear Velocty Lnear velocty - The nstantaneous rate of change n lnear poston of a pont relatve to some frame. 4

6 Defntons - Lnear Velocty The poston of pont Q n frame {A} s represented by the lnear poston vector The velocty of a pont Q relatve to frame {A} s represented by the lnear velocty vector 5

7 Defntons - Angular Velocty - Vector Angular Velocty Vector: A vector whose drecton s the nstantaneous axs of rotaton of one frame relatve to another and whose magntude s the rate of rotaton about that axs. 6

8 Defntons - Angular Velocty Angular Velocty: The nstantaneous rate of change n the orentaton of one frame relatve to another. 7

9 Defntons - Angular Velocty Just as there are many ways to represent orentaton (Euler Angles, Roll-Ptch-Yaw Angles, Rotaton Matrces, etc.) there are also many ways to represent the rate of change n orentaton. The angular velocty vector s convenent to use because t has an easy to grasp physcal meanng. However, the matrx form s useful when performng algebrac manpulatons. 8

10 Lnear & Angular Veloctes - Frames When descrbng the velocty (lnear or angular) of an object, there are two mportant frames that are beng used: Represented Frame (Reference Frame) : Ths s the frame used to represent (express) the object s velocty. Computed Frame Ths s the frame n whch the velocty s measured (dfferentate the poston). 9

11 Frame - Velocty As wth any vector, a velocty vector may be descrbed n terms of any frame, and ths frame of reference s noted wth a leadng superscrpt. A velocty vector computed n frame {B} and represented n frame {A} would be wrtten Represented (Reference Frame) Computed (Measured) 10

12 Lnear Velocty - Rgd Body Gven: Consder a frame {B} attached to a rgd body whereas frame {A} s fxed. The orentaton of frame {A} wth respect to frame {B} s not changng as a functon of tme Problem: descrbe the moton of of the vector relatve to frame {A} 11 Soluton: Frame {B} s located relatve to frame {A} by a poston vector and the rotaton matrx (assume that the orentaton s not changng n tme ) expressng both components of the velocty n terms of frame {A} gves

13 12 Lnear Velocty - Rgd Body

14 Angular Velocty - Rgd Body Gven: Consder a frame {B} attached to a rgd body whereas frame {A} s fxed. The vector s constant as vew from frame {A} Problem: descrbe the velocty of the vector representng the pont Q relatve to frame {A} Soluton: Even though the vector s constant as vew from frame {B} t s clear that pont Q wll have a velocty as seen from frame {A} due to the rotatonal velocty 13

15 14 Angular Velocty - Rgd Body - Intutve Approach

16 15 Angular Velocty - Rgd Body - Intutve Approach

17 16 Angular Velocty - Rgd Body - Intutve Approach

18 17 Angular Velocty - Rgd Body - Intutve Approach

19 18 Angular Velocty - Rgd Body - Intutve Approach

20 19 Angular Velocty - Rgd Body - Intutve Approach

21 Angular Velocty - Rgd Body - Intutve Approach The fgure shows to nstants of tme as the vector rotates around. Ths s what an observer n frame {A} would observe. The Magntude of the dfferental change s Usng a vector cross product we get 20

22 Angular Velocty - Rgd Body - Intutve Approach In the general case, the vector Q may also be changng wth respect to the frame {B}. Addng ths component we get. Usng the rotaton matrx to remove the dual-superscrpt, and snce the descrpton of at any nstance s we get 21

23 MORE ON ANGULAR VELOCITY A property of the dervatve of an orthonormal matrx R SR 22 Note: S s called the skew-symmetrc matrx.

24 Velocty of a pont due to rotatng reference frame B P constant or 23

25 Skew-symmetrc matrces and the vector cross product angular velocty matrx angular velocty vector Matrx Form Vector Form 24

26 Ganng physcal nsght concernng the angular velocty vector Rotatng about the fxed frame: operator kkv xxc kkv xyks z kkv xzks y RK() kkv x y ks z kkv y y c kkv y z ks x kkv xzks y kkv yzks x kkv zzc A ssn, ccos, vs1cos Kˆ [ k, k, k ] x y z T (2.80) 25

27 Ganng physcal nsght concernng the angular velocty vector lm ( ) z y z x y x t k k k k k k R Rt t 0 0 () 0 z y z x y x k k R k k Rt k k 1 ( ) 1 1 z y K z x y x k k R k k k k 3 0 ( ) lm ( ) k t R I R R t t z y z x y x k k RR k k k k 26

28 R Ganng physcal nsght concernng the angular velocty vector 0 k z k y 0 z y 1 RR k z 0 k xz 0 xs k y k x 0 y x 0 x k x SR ˆ y k y K z k z angular velocty matrx angular velocty vector The physcal meanng of the angular-velocty vector s that, at any nstant, the change n orentaton of a rotatng frame can be vewed as a rotaton about some axs. ˆK Ths nstantaneous axs of rotaton, taken as a unt vector and then scaled by the speed of rotaton about the axs ( ), yelds the angular-velocty vector. 27

29 Angular Velocty - Matrx & Vector Forms S= 28

30 Smultaneous Lnear and Rotatonal Velocty The fnal results for the dervatve of a vector n a movng frame (lnear and rotaton veloctes) as seen from a statonary frame Vector Form Matrx Form 29

31 Smultaneous Lnear and Rotatonal Velocty - Vector Versus Matrx Representaton Vector Form Matrx Form 30

32 Poston Propagaton The homogeneous transform matrx provdes a complete descrpton of the lnear and angular poston relatonshp between adjacent lnks. These descrptons may be combned together to descrbe the poston of a lnk relatve to the robot base frame {0}. A smlar descrpton of the lnear and angular veloctes between adjacent lnks as well as the base frame would also be useful. 31

33 Moton of the Lnk of a Robot In consderng the moton of a robot lnk we wll always use lnk frame {0} as the reference frame Where: - s the lnear velocty of the orgn of lnk frame () wth respect to frame {0} - s the angular velocty of the orgn of lnk frame () wth respect to frame {0} 32

34 Frame Velocty-revew As wth any vector, a velocty vector may be descrbed n terms of any frame, and ths frame of reference s noted wth a leadng superscrpt. A velocty vector computed n frame {B} and represented n frame {A} would be wrtten Represented (Reference Frame) Computed (Measured) 33

35 Veloctes - Frame & Notaton Expressng the velocty of a frame {} (assocated wth lnk ) relatve to the robot base (frame {0}) usng our prevous notaton s defned as follows: The veloctes dfferentate (computed) relatve to the base frame {0} are often represented relatve to other frames {k}. The followng notaton s used for ths condtons 34

36 Velocty Propagaton Gven: A manpulator - A chan of rgd bodes each one capable of movng relatve to ts neghbour Problem: Calculate the lnear and angular veloctes of the lnk of a robot Soluton (Concept): Due to the robot structure we can compute the veloctes of each lnk n order startng from the base. 35 The velocty of lnk +1 wll be that of lnk, plus whatever new velocty components were added by jont +1

37 Velocty of Adjacent Lnks - Angular Velocty 1/5 From the relatonshp developed prevously we can re-assgn lnk names to calculate the velocty of an lnk relatve to the base frame {0} By pre-multplyng both sdes of the equaton by,we can convert the frame of reference for the base {0} to frame {+1} 36

38 Velocty of Adjacent Lnks - Angular Velocty 2/5 Usng the recently defned notaton, we have -Angular velocty of frame {+1} measured relatve to the robot base, and expressed n frame {+1} -Angular velocty of frame {} measured relatve to the robot base, and expressed n frame {+1} -Angular velocty of frame {+1} measured relatve to frame {} and expressed n frame {+1} 37

39 Velocty of Adjacent Lnks - Angular Velocty 3/5 Angular velocty of frame {} measured relatve to the robot base, expressed n frame {+1} In the second term Angular velocty of frame {+1} measured (dfferentate) n frame {} and represented (expressed) n frame {+1} s shown n the second term 38

40 Velocty of Adjacent Lnks - Angular Velocty 4/5 Assumng that a jont has only 1 DOF. The jont confguraton can be ether revolute jont (angular velocty) or prsmatc jont (Lnear velocty). Based on the frame attachment conventon n whch we assgn the Z axs pontng along the +1 jont axs such that the two are concde (rotatons of a lnk s preformed only along ts Z- axs) we can rewrte ths term as follows: 39

41 Velocty of Adjacent Lnks - Angular Velocty 5/5 The result s a recursve equaton that shows the angular velocty of one lnk n terms of the angular velocty of the prevous lnk plus the relatve moton of the two lnks. Snce the term depends on all prevous lnks through ths recurson, the angular velocty s sad to propagate from the base to subsequent lnks. 40

42 Velocty of Adjacent Lnks - Lnear Velocty 1/5 Smultaneous Lnear and Rotatonal Velocty The dervatve of a vector n a movng frame (lnear and rotaton veloctes) as seen from a statonary frame Vector Form Matrx Form 41

43 Velocty of Adjacent Lnks - Lnear Velocty 2/5 From the relatonshp developed prevously (matrx form) we re-assgn lnk frames for adjacent lnks ( and +1) wth the velocty computed relatve to the robot base frame {0} Q By pre-multplyng both sdes of the equaton by, we can convert the frame of reference for the left sde to frame {+1} 42

44 Velocty of Adjacent Lnks - Lnear Velocty 3/5 -Lnear velocty of frame {+1} measured relatve to frame {} and expressed n frame {+1} Assumng that a jont has only 1 DOF. The jont confguraton can be ether revolute jont (angular velocty) or prsmatc jont (Lnear velocty). Based on the frame attachment conventon n whch we assgn the Z axs pontng along the +1 jont axs such that the two are concde (translaton of a lnk s preformed only along ts Z- axs) we can rewrte ths term as follows: 43

45 Velocty of Adjacent Lnks - Lnear Velocty 4/5 Wth replacement: Defnton 44

46 Angular Velocty - Matrx & Vector Forms S= 45

47 Velocty of Adjacent Lnks - Lnear Velocty 5/5 The result s a recursve equaton that shows the lnear velocty of one lnk n terms of the prevous lnk plus the relatve moton of the two lnks. Snce the term depends on all prevous lnks through ths recurson, the angular velocty s sad to propagate from the base to subsequent lnks. 46

48 Velocty of Adjacent Lnks - Summary Angular Velocty 0 - Prsmatc Jont Lnear Velocty 0 - Revolute Jont 47

49 Example-2R Robot Fnd and 48

50 49 Example-2R Robot

51 Example-2R Robot For =0 = =0 50

52 For Example-2R Robot =0 51

53 Example-2R Robot For =0 or 52

54 Example-2R Robot or 53

55 Angular and Lnear Veloctes - 3R Robot - Example For the manpulator shown n the fgure, compute the angular and lnear velocty of the tool frame relatve to the base frame expressed n the tool frame (that s, calculate 54

56 Angular and Lnear Veloctes - 3R Robot - Example Frame attachment 55

57 Angular and Lnear Veloctes - 3R Robot - Example DH Parameters 56

58 Angular and Lnear Veloctes - 3R Robot - Example From the DH parameter table, we can specfy the homogeneous transform matrx for each adjacent lnk par: 0 1R 57

59 Angular and Lnear Veloctes - 3R Robot - Example Compute the angular velocty of the end effector frame relatve to the base frame expressed at the end effector frame. For =0 1 0 R 0 1 R T 58

60 59 Angular and Lnear Veloctes - 3R Robot - Example

61 Angular and Lnear Veloctes - 3R Robot - Example Compute the lnear velocty of the end effector frame relatve to the base frame expressed at the end effector frame. Note that the term nvolvng the prsmatc jont has been dropped from the equaton (t s equal to zero). 60

62 Angular and Lnear Veloctes - 3R Robot - Example For =0 For =1 61

63 62 Angular and Lnear Veloctes - 3R Robot - Example

64 Angular and Lnear Veloctes - 3R Robot - Example 0 Ls Lc 2 3 L3 L 3 2 J( ) L L c L c

65 Angular and Lnear Veloctes - 3R Robot - Example Note that the lnear and angular veloctes ( ) of the end effector where dfferentate (measured) n frame {0} however represented (expressed) n frame {4} In the car example: Observer sttng n the Car Observer sttng n the World 64

66 Angular and Lnear Veloctes - 3R Robot - Example Multply both sdes of the equaton by the nverse transformaton matrx, we fnally get the lnear and angular veloctes expressed and measured n the statonary frame {0} 65

67 Knematcs Relatons - Jont & Cartesan Spaces A robot s often used to manpulate object attached to ts tp (end effector). The locaton of the robot tp may be specfed usng one of the followng descrptons: Jont Space Cartesan Space 66 Euler Angles

68 Knematcs Relatons - Forward & Inverse The robot knematc equatons relate the two descrpton of the robot tp locaton Tp Locaton n Jont Space Tp Locaton n Cartesan Space 67

69 Knematcs Relatons - Forward & Inverse Tp Velocty n Jont Space Tp velocty n Cartesan Space 68

70 Jacoban Matrx - Introducton The Jacoban s a mult dmensonal form of the dervatve. Suppose that for example we have 6 functons, each of whch s a functon of 6 ndependent varables We may also use a vector notaton to wrte these equatons as 69

71 Jacoban Matrx - Introducton If we wsh to calculate the dfferental of as a functon of the dfferental we use the chan rule to get Whch agan mght be wrtten more smply usng a vector notaton as 70

72 Jacoban Matrx - Introducton The 6x6 matrx of partal dervatve s defned as the Jacoban matrx By dvdng both sdes by the dfferental tme element, we can thnk of the Jacoban as mappng veloctes n X to those n Y Note that the Jacoban s tme varyng lnear transformaton 71

73 Jacoban Matrx - Introducton In the feld of robotcs the Jacoban matrx descrbe the relatonshp between the jont angle rates ( ) and the translaton and rotaton veloctes of the end effector ( ). Ths relatonshp s gven by: 72

74 Jacoban Matrx - Introducton Ths expresson can be expanded to: Where: - s a 6x1 vector of the end effector lnear and angular veloctes s a 6xN Jacoban matrx s a Nx1 vector of the manpulator jont veloctes N s the number of jonts 73

75 How to obtan the Jacoban for a gven robot? Accordng to defnton of the Jacoban s gven n the Eqs. (5.58) through (5.62), a Jacoban can be obtaned by drectly dfferentatng the knematc equatons of the mechansm of the gven robot..e., The knematc equatons of a robot can be obtaned va the geometrc approach or algebrac approach (chapter three). However, whle ths s straghtforward for lnear velocty, there s no 3 1 orentaton vector whose dervatve s. 74

76 How to obtan the Jacoban for a gven robot? Alternatvely, we can use the velocty propagaton method to derve the Jacoban usng successve applcaton of the recursve equaton For revolute jont: For prsmatc jont: 75

77 76 Jacoban Matrx - Calculaton Methods

78 Jacoban Matrx by Dfferentaton - 3R - 1/ ) sn( ) sn( sn ) cos( ) cos( cos L L L y L L L x Consder the followng 3 DOF Planar manpulator (Geometrc approach) 77

79 Jacoban Matrx by Dfferentaton - 3R - 2/3 (Geometrc approach) The forward knematcs gves us relatonshp of the end effector to the jont angles: Dfferentatng the three expressons gves 78

80 Jacoban Matrx by Dfferentaton - 3R - 3/3 Usng a matrx form we get 79 The Jacoban provdes a lnear transformaton, gvng a velocty map and a force map for a robot manpulator. For the smple example above, the equatons are trval, but can easly become more complcated wth robots that have addtonal degrees a freedom. Before tacklng these problems, consder ths bref revew of lnear algebra.

81 Jacoban: Velocty propagaton The recursve expressons for the adjacent jont lnear and angular veloctes descrbe a relatonshp between the jont angle rates ( ) and the transnatonal X and rotatonal veloctes of the end effector ( ): 80

82 Jacoban: Velocty propagaton Therefore the recursve expressons for the adjacent jont lnear and angular veloctes can be used to determne the Jacoban n the end effector frame Ths equaton can be expanded to: 81

83 Angular and Lnear Veloctes - 3R Robot - Example For the manpulator shown n the fgure, compute the angular and lnear velocty of the tool frame relatve to the base frame expressed n the tool frame (that s, calculate 82

84 Angular and Lnear Veloctes - 3R Robot - Example Compute the angular velocty of the end effector frame relatve to the base frame expressed at the end effector frame. For =0 1 0 R 0 1 R T 83

85 Angular and Lnear Veloctes - 3R Robot - Example 0 Ls Lc 2 3 L3 L 3 2 J( ) L L c L c

86 Jacoban: Veloctes and sngulartes Mappng from veloctes n the jont space (jont) to veloctes n the Cartesan space (end effector) At certan ponts, called sngulartes, ths mappng s not nvertble. 85

87 Sngularty - The Concept Motvaton: We would lke the hand of a robot (end effecror) to move wth a certan velocty vector n Cartesan space. Usng lnear transformaton relatng the jont velocty to the Cartesan velocty we could calculate the necessary jont rates at each nstance along the path. Gven: a lnear transformaton relatng the jont velocty to the Cartesan velocty (usually the end effector) Queston: Is the Jacoban matrx nvertble? (Or) Is t nonsngular? Is the Jacoban nvertble for all values of? If not, where s t not nvertble? 86

88 How to fnd the sngulartes? The roots of the followng equatons determne the sngulartes: where Det[J ( )] s the determnant of the Jacoban under consderaton. Certanly, a nonsquare Jacoban s not nvertble. However, even f a Jacoban s n a square matrx format, the determnant of the Jacoban may stll be zero. 87

89 What s the geometrc or physcal meanngs of sngulartes? Cleary, when a robot s n a sngular confguraton, t has lost one or more degree of freedom as vewed from Cartesan space. Ths means that there s some drecton (or subspace) n Cartesan space along whch t s mpossble to move the hand of the robot no matter whch jont rates are selected. It s obvous ths happens at the workspace boundary of robots. 88

90 Where do sngulartes occur? All mechansms have sngulartes at the boundary of ther workspace, and most have loc of sngulartes nsde ther workspace. Hence sngulartes can be classfed nto two categores: 1. Workspace boundary sngulartes are those whch occur when the manpulator s fully stretched out or folded back on tself such that the end-effector s near or at the boundary of the workspace. Vewed from Cartesan space Workspace nteror sngulartes are those whch occur away from the workspace boundary and generally are caused by two or more jont axes lnng up, or due to the mechansm structure so that Det[J ( )] = f ( ) = 0.

91 More on sngularty Snce dfferent structures of the robot yelds dfferent lnk parameters, therefore, Det[J ( )] = f ( ) = 0 occur n the dfferent places. By modfyng the lnk parameters, one can elmnate the sngulartes nsde the workspace. Note: Det[J ( )] = f ( ) = 0 provdes only one equatons. For f ( ) = 0, - f there s only one varable nsde the equaton, the sngularty s only at ndvdual ponts; - f there are two varables nsde the equaton, the sngulartes occur alone a lne; - f more than three varables, the sngulartes form a hypersurface. 90

92 Sngularty -Summary 91 Losng one or more DOF means that there s a some drecton (or subspace) n Cartesan space along whch t s mpossble to move the hand of the robot (end effector) no matter whch jont rate are selected

93 Sngularty -Summary Answer (Conceptual): Most manpulator have values of where the Jacoban becomes sngular. Such locatons are called sngulartes of the mechansm or sngulartes for short 92

94 Propertes of the Jacoban - Velocty Mappng and Sngulartes Where are the sngulartes? What s the physcal explanaton of the sngulartes? Are they workspace boundary or nteror sngulartes? From example

95 Propertes of the Jacoban - Velocty Mappng and Sngulartes They are workspace boundary sngulartes because they exst at the edge of the manpulator workspace. At both case, the moton of the end-effector s possble only along one Cartesan drecton (the one perpendcular to the arm). Therefore, the mechansm has lost one degree of freedom. At sngularty confguraton, the nverse Jacoban blows up! Ths results n jont rates approachng nfnty as the sngularty s approached. It s very dangers n a robot control system. 94

96 95 Propertes of the Jacoban - Velocty Mappng and Sngulartes

97 Propertes of the Jacoban - Velocty Mappng and Sngulartes Example: Planar 3R 96

98 Propertes of the Jacoban - Velocty Mappng and Sngulartes The manpulator loses 1 DEF. The end effector can only move along the tangent drecton of the arm. Moton along the radal drecton s not possble. 97

99 Angular and Lnear Veloctes - 3R Robot - Example For the manpulator shown n the fgure, compute the angular and lnear velocty of the tool frame relatve to the base frame expressed n the tool frame (that s, calculate Workspace Interor Sngulartes? 98

100 Angular and Lnear Veloctes - 3R Robot - Example Det( 0 V 4 ) Det( 0 4 R 0 Ls 0 L1Lc 2 2Lc Lc 2 3 L3 L 3 2 J( ) 4 det( J ( )) ( L`1 L2c2 L3c23 ) L2L3s3 L Lc Lc L Ls 0 `1 4 V 4 2 ) 2 Det( R) Det( J ( ) ) det( 4 J( )) Draw the locus of the sngular ponts of the robot n ts workspace (or n ts base frame). (Interor sngulartes?)

101 Angular and Lnear Veloctes - 3R Robot - Example dt dp J V vew sde L L Z vew top L L L Y vew top L L L X Tp Tp Tp Tp / ) ( ) sn( sn sn )) cos( cos ( cos )) cos( cos ( How to fnd? ) ( J or V Geometrc approach: 100

102 Jacoban: Veloctes and Statc Forces Mappng from veloctes n the jont space (jont) to veloctes n the Cartesan space (end effector) Mappng from the jont force/torques to forces/torque n the Cartesan space appled on the end effector - F 101

103 Statc Analyss Protocol - Free Body Dagram 1/ Step 1 Lock all the jonts - Convertng the manpulator (mechansm) to a structure Step 2 Consder each lnk n the structure as a free body and wrte the force / moment equlbrum equatons 102 Step 3 Solve the equatons - 6 Eq. for each lnk. Apply backward soluton startng from the last lnk (end effector) and end up at the frst lnk (base)

104 Statc Analyss Protocol - Free Body Dagram 2/ Specal Symbols are defned for the force and torque exerted by the neghbor lnk - Force exerted on lnk by lnk -1 - Torque exerted on lnk by lnk -1 Reference coordnate system {B} Force f or torque n Exerted on lnk A by lnk A-1 For easy soluton superscrpt ndex (B) should the same as the subscrpt (A) 103

105 Statc Analyss Protocol - Free Body Dagram 3/ For seral manpulator n statc equlbrum (jonts locked), the sum the forces and torques actng on lnk n the lnk frame {} are equal to zero. 104

106 Statc Analyss Protocol - Free Body Dagram 4/ Procedural Note: The soluton starts at the end effector and ends at the base Re-wrtng these equatons n order such that the known forces (or torques) are on the rght-hand sde and the unknown forces (or torques) are on the left, we fnd 105

107 Statc Analyss Protocol - Free Body Dagram 5/ Changng the reference frame such that each force (and torque) s expressed upon ther lnk s frame, we fnd the statc force (and torque) propagaton from lnk +1 to lnk These equatons provde the statc force (and torque) propagaton from lnk to lnk. They allow us to start wth the force and torque appled at the end effector, and calculate the force and torque at each jont all the way back to the robot base frame. 106

108 Statc Analyss Protocol - Free Body Dagram 6/ Queston: What torques are needed at the jonts n order to balance the reacton forces and moments actng on the lnk. Answer: All the components of the force and moment vectors are ressted by the structure of mechansm tself, except for the torque about the the jont axs. Therefore, to fnd the jont the torque requred to mantan the statc equlbrum, the dot product of the jont axs vector wth the moment vector actng on the lnk s computed Revolute Jont Prsmatc Jont 107

109 Example - 2R Robot - Statc Analyss Problem Gven: -2R Robot -A Force vector s appled by the end effector Compute: The requred jont torque as a functon of the robot confguraton and the appled force 108

110 Example - 2R Robot - Statc Analyss Soluton Lock the revolute jonts Apply the statc equlbrum equatons startng from the end effector and gong toward the base 109

111 Example - 2R Robot - Statc Analyss For =2 110 Vector cross product

112 Example - 2R Robot - Statc Analyss For =1 111

113 112 Example - 2R Robot - Statc Analyss

114 Example - 2R Robot - Statc Analyss Re-wrtng the equatons n a matrx form 113

115 Jacobans n the force doman We have found jont torques that wll exactly balance forces at the end-effector n the statc stuaton. When force act on a mechansm, work (n the techncal sense) s done f the mechansm moves through a dsplacement. Work s defned as a force actng through a dstance and s a scalar wth unts of energy. The prncple of vrtual work allows us to make certan statement about the statc case by allowng the amount of ths dsplacement to go to an nfntesmal. Note : The unts of energy are nvarant after coordnate changng. 114

116 The prncple of vrtual work In the multdmensonal case, work s the dot product of a vector force or torque and a vector dsplacement: F s a 6 1 Cartesan force-torque vector actng at the end-effctor, X s a 6 1 nfntesmal Cartesan dsplacement of the end-effector, s a 6 1 vector of torques at the jonts of the robot, s a 6 1 vector of nfntesmal jont dsplacements. Inner product can also be expressed as: where t must hold for all, so we have 115

117 Jacobans: Veloctes and statc forces For velocty: For force-torque: If the Jacoban s wth respect to frame {0}, we have When the Jacoban loses full rank, there are certan drectons n whch the end-effector cannot exert statc forces as desred. 116

118 Jacobans n the force doman If the Jacoban s sngular, F could be ncreased or decreased n certan drectons (those defnng the null-space of the Jacoban) wth no effect on the value calculated for. Ths also means that near sngular confguratons, mechancal advatage tends towards nfnty such that wth small jont torques large forces could be generated at the end-effector. 117

119 Propertes of the Jacoban - Force Mappng and Sngulartes The relatonshp between jont torque and end effector force and moments s gven by: The rank of s equals the rank of. At a sngular confguraton there exsts a non trval force such that In other words, a fnte force can be appled to the end effector that produces no torque at the robot s jonts. In the sngular confguraton, the manpulator can lock up. 118

120 Propertes of the Jacoban - Force Mappng and Sngulartes Ths stuaton s an old and famous one n mechancal engneerng. For example, n the steam locomotve, top dead center refers to the followng condton The pston force, F, cannot generate any torque around the drve wheel axs because the lnkage s sngular n the poston shown. 119

121 120 Cartesan transformaton of veloctes and statc forces

122 Cartesan transformaton of veloctes and statc forces If the frames {A} and {B} are rgdly connected, the matrx operator form to transform general velocty vectors n frame {A} to ther descrpton n frame {B} s gven as follows: It can be shown by substtutng nto equatons (5.45) and (5.47) and re-expressed the two equatons n matrx form. v R R P R v ? 121

123 Cartesan transformaton of veloctes and statc forces By defnng a 6 6 operator: a velocty transformaton, whch maps veloctes n {A} nto velcotes n {B}, we have The nverse transformaton whch maps veloctes n {B} to {A} s gven n the smlar matrx form as or R P v R P v R v Note that R ( 1) P v R v 122

124 Cartesan transformaton of veloctes and statc forces where s used to denote a force-momonet transformaton. Smlarly, wth the statc forces relatons We have the n f R R P R n f 123

125 Cartesan transformaton of veloctes and statc forces Velocty and force transformatons are smlar to Jacobans n that they related velcottes and force n dfferent coordnate systems. Smlarly to Jacobans we have that as can be verfed by examnng by the followng equatons 124

126 Example Gven: -the output of the force Sensor. -the transformaton related the tool frame to the sensor frame. Fnd: Soluton: where 125